where the Hamiltonian \(H : {\mathbb {R}}^d\times {\mathbb {R}}^d \times {\mathbb {R}}^{d\times d} \rightarrow {\mathbb {R}}\cup \lbrace \infty \rbrace \)  is a lower semi-continuous convex function with respect to the two last arguments \((z,\gamma )\) ,
and \(g\)  a measurable function on \({\mathbb {R}}^d\) . The numerical resolution of this class of PDE is a notorious challenging problem, and it is especially difficult to obtain a good approximation of the second spatial derivative \(D_x^2 u\)  of the solution in this fully nonlinear context. In the last years, significant progress has been achieved towards these challenges with several numerical methods using techniques from deep learning, see the recent surveys by [1]} and [2]}:
A first class of approximation algorithms, called Physics Informed Neural Network (PINNs) [3]}, also known as Deep Galerkin method (DGM) [4]}, directly approximates the solution to the PDE by a neural network, and its partial derivatives by automatic differentiation, by minimizing the loss function arising from the residual of the PDE evaluated on a random grid in the space-time domain. A second class of algorithms relies on the backward stochastic differential representation of
the PDE in the semi-linear case by minimizing either a global loss function (see [5]}, and extensions in
[6]}, [7]}, [8]}), or sequence of loss functions from backward recursion (see [9]}, and variations-extensions in [10]}, [11]}, [2]}).
